Problem: How many numbers between $1$ and $100$ (inclusive) are divisible by $4$ or $5$ ?
Answer: There are $25$ numbers divisible by $4$ between $1$ and $100$, and $20$ numbers divisible by $5$ between $1$ and $100$. So, you might think there are $25 + 20 = 45$ numbers divisible by one or the other, but this is overcounting something. We're counting every number which is divisible by both $4$ and $5$ twice. So, for example, $20$ is counted once as a number divisible by $4$, and then again as a number divisible by $5$. So, we need to count how many numbers are divisible by both $4$ and $5$ and subtract this from what we had before. Being divisible by both $4$ and $5$ is the same thing as being divisible by $20$, so there are $5$ numbers between $1$ and $100$ divisible by both. Subtracting, there are $45 - 5 = 40$ numbers divisible by $4$ or $5$.